Because LOCC is just a classical process in communication, entanglement between A and B is defined as a quantum effect such that the amount of entanglement never increases by LOCC. Besides, a product state of pure states has minimum entanglement and the value is zero. This definition leads to a fact that any separable state has zero entanglement. ( ) + + ⊗ + + = 0 ENT AB A A B B ∞ ( ) ( ) ∑ µ ρ µ ⊗ ρ µ = ( ) 0 . ENT p AB A B µ = 1 Entangled states are defined as non-seperable states: ∞ ≠ ∑ ( ) ( ) ρ µ ρ µ ⊗ ρ µ > ( ) 0 ENT p AB AB A B µ = 1
Increasing entanglement requires non-local operations directly connecting A to B, or quantum channels which can transport not only classical but also quantum information (quantum states). Non-Local Operation A B interaction Quantum Channel Ψ B A 0 , 1 Entanglement is a purely quantum effect and generates interesting phenomena, including break of Bell’s inequality and quantum teleportation.
Example: entanglement of pure state of two systems Contracted State: [ ] [ ] ρ = Ψ Ψ ρ = Ψ Ψ Tr , Tr A B AB AB B A AB AB Entanglement Entropy as one of entanglement indices [ ] [ ] Ψ Ψ = − ρ ρ = − ρ ρ ( ) Tr ln Tr ln EE AB AB AB A A A B B B
Schmidt Decomposition of Pure State: Ψ = + p u v p u v Ψ + + + Ψ − − − AB A B A B ≥ ∑ = = δ = δ 0 , 1 , , p p u u v v Ψ ± Ψ ' ' ' ' s As As ss Bs Vs ss = ± s ( ) ∑ ⇒ Ψ Ψ = − ln EE p p Ψ Ψ AB AB AB s s = ± s The Bell states attain the maximum value of entanglement entropy. [ ] 1 = + + + − − Bell 2 ( ) ⇒ = ln 2 EE AB Bell Bell
II. Lubkin-Lloyd-Pagels-Page Theorem, Page Curve Hypothesis and BH Firewall Conjecture
The BH firewall conjecture is based on the Page curve hypothesis, and the hypothesis was inspired by the Lubkin-Lloyd-Pagels-Page (LLPP) theorem.
Lubkin-Lloyd-Pagels-Page Theorem: Typical states of A and B are almost maximally entangled when the systems are large. N A B N A Ψ B AB N << N Typical State of AB A B [ ] ρ = Ψ Ψ Tr Ψ AB A B AB AB [ ] = − ρ ln ρ S Tr EE A A A 1 ≈ ρ ≈ ln I S N A A EE A N A
Maximal Entanglement between A and B N 1 1 ∑ A ~ = ρ = ⇒ Max u v I A A n n AB A B N N = 1 n A A Orthogonal unit vectors N ≥ N B A
Let us assume that Hilbert-space dimensions of black holes and Hawking radiation become finite due to quantum gravity effect. Page’s Strategy for Finding States of BH Evaporation: Nobody knows exact quantum gravity dynamics. So let’s gamble that the state scrambled by quantum gravity is one of TYPICAL pure states of the finite- dimensional composite system! That may not be so bad!
Simplified Page Curve S EE Maximum value of EE is attained at each time. HR BH = dim , BH H BH OLD BH = dim HR H ln HR HR <<Page Time>> ≈ ln ln BH HR ○ ≈ 0 . 7 M M ○ page bh A << << ⇒ ≈ = BH 1 ln BH HR S BH EE 4 G
Page Curve Hypothesis for BH Evaporation: Proposition I : When the dimension of the BH Hilbert space is much larger or less than that of Hawking radiation, BH and HR in a typical pure state of quantum gravity share almost maximal entanglement. In other words, quantum states of the smaller system is almost proportional to the unit matrix. Proposition II : = S S of the smaller system. EE thermal
OLD BH Late radiation B Early radiation C A Ψ = ∪ HR A B ABC BH = C Page Time 1 << , , A B C << B C A OLD BH ⇒
Proposition I means that A and BC are almost maximally entangled with each other. B C A 1 1 1 ρ = = ⊗ I I I BC BC B C BC B C NO CORRELATION BETWEEN B AND C! Harrow-Hayden
B 1 1 C ρ = ⊗ A I I BC B C B C x C ε x B 2 ϕ − ϕ ( ) ( ) 1 x x ρ = B C Tr O ε ε BC 2 ε → 0 [ ] ( ) ∂ ϕ ρ = ∞ 2 ( ) Tr x BC FIREWALL!
III. Canonical Typicality for Non-Vanishing Hamiltonian, and No Firewalls
Problem for Proposition I of Page Curve Hypothesis: The area law of entanglement entropy is broken in a sense of ordinary many body physics, though outside-horizon energy density in BH evaporation is much less than the Planck scale.
∝ ∂ = ∂ ← standard area law of | | | | S EE A B entanglement entropy A Ψ ≈ 0 AB AB B for low excited states
| | A 1 ∑ ~ Ψ = u v LLPP Typicality ⇒ n n AB A B | | A = 1 n Qubit network model A = V 2 A A = ln ∝ S A V B EE A Not area law, but volume law for highly excited states!
This is because zero Hamiltonian (complete degeneracy) is assumed in the LLPP theorem. This is also an implicit premise of the Page curve hypothesis. = 0 . H AB
In BH physics, we have to treat canonical typicality with non-vanishing H in a precise manner. Then non-maximal entanglement emerges and makes near-horizon regions smooth. Thus no firewalls appear. M. Hotta and A. Sugita, Prog. Theor. Exp. Phys, 123B04 (2015).
Microcanonical Energy Shell （ not a tensor product of the sub-Hilbert spaces ） = H E E E AB j j j { } [ ] ∆ = ∈ − δ ( ) | , E j E E E j = ∑ Microcanonical ( ) V E c E ES j j Energy Shell: ∈ ∆ ( ) j E
Ψ AB ∈ A B ( E ) V ES N N A B [ ] ρ = Ψ Ψ Tr A B AB AB 2 − 1 N 1 ∑ A = + I T T A n n N = 1 n A Bloch Representation of higher-dim quantum states [ ] [ ] = = = δ † , 0 , T T Tr T Tr T T N n ' ' n n n n A nn [ ] = ρ T Tr T n n A
∑ Ψ = . c E T Evaluate for j j n AB ∈ ∆ ( ) j E ∑ = = dim ( ) dim D V E c E ES j j ∈ ∆ ( ) j E N << N Volume of B A B Hilbert space dimension of B ∝ γ >> exp( ( )) 1 D V B N B for ordinary systems.
Uniform Ensemble on Mircrocanonical Energy Shell: ∫ = ∑ D 2 ( ) 1 ∝ δ − p c d p ( ) 1 p c c j ∈ ∆ ( ) j E ∫ = D ( ) ( ) f f c p c d p 1 ∗ = δ , c c ' ' j j jj D ( ) 1 ∗ ∗ = δ δ + δ δ c c c c + ' ' ' ' ' ' j k j k jj kk jk j k ( 1 ) D D
( ) 2 T Max eigenvalue 2 n − ≤ T T + n n 1 D ( ) 2 − 1 N 1 1 ∑ A 2 ρ − ρ ≤ 2 Tr T + A A A n 1 N D = 1 n A N independent! B ( ( ) ) ∝ γ = >> 23 exp( ) exp 10 1 D V O B Sugita Theorem (2006) ρ − ρ ≤ − γ (exp( )) O V A A B
Hotta-Sugita (2015) as a response to a BH firewall debate with Daniel Harlow Private Communication with D. Harlow about “Jerusalem Lectures on Black Holes and Quantum Information”, arXiv:1409.1231 .
Harlow argued a canonical typicality in a weak interaction limit. = + + H H H A B Negligibly small = + = . H E E const A B
+ = . E E const A B N >> A B N B A Ψ AB [ ] ρ = Ψ Ψ Tr Ψ A B AB AB AB [ ] = − ρ ln ρ S Tr AB A A A 1 ( ) ρ ≈ − β exp H A A Z A Without any proof, Harlow ≈ β ( ) S S argued these only in the weak , AB thermal A interaction limit.
Harlow pointed out a possibility that BH firewalls may exist even after canonical typicality with non-zero Hamiltonian. B BH = = ∪ C , HR A B C A 1 << << , , A B C B C A ( ( ) ) ρ ∝ − β + exp H H BC B C ( ) ( ) = − β ⊗ − β exp exp H H B C I ⊗ I No Correlation, just like B C [ ] ( ) ∂ ϕ ρ = ∞ 2 ( ) ? Tr x BC FIREWALL? Harlow, arXiv:1409.1231
However, the worry is useless. We can prove nonexistence of firewalls for general systems by using the general theory of canonical typicality. M. Hotta and A. Sugita, Prog. Theor. Exp. Phys, 123B04 (2015). Irrespective of the strength of the interaction between B and C, ( ( ) ) ρ ∝ − β + + exp H H V BC B C BC << B C A
Actually, a correlation exists between B and C for small interactions. V BC H H C B ( ( ) ) ρ ∝ − β + + exp H H V BC B C BC [ ] Tr ρ = ∞ lim ! ? BC V Harlow’s worry: BC → 0 V BC
Border shift does not change physics at all. + + = + + ' ' ' H H V H H V ' ' ' ' B C BC B C B C ( ( ) ) ρ = ρ ∝ − β + + ' exp ' ' ' H H V ' ' ' ' ' ' BC B C B C B C V ' V BC ' C ' B ' H ' H ' ' C B Merely an ordinary local operator of C’ [ ] [ ] ρ = ρ < ∞ No firewall! ' Tr V Tr V ' ' BC BC B C BC
Remark: for ordinary weakly interacting quantum systems, entanglement entropy is upper bounded by thermal entropy, as long as stable Gibbs states exist. H H B A + = A B E E E A B Ψ AB [ ] ρ = Ψ Ψ Tr Arbitrary state: A B AB AB ρ = − β β exp( ( ) ) / ( ( )) E H Z E Gibbs state: A A A [ ] [ ] = − ρ ρ ≤ − ρ ρ = ln ln S Tr Tr S EE A A A A thermal
Conventional “proof”: [ ] [ ] [ ] ( ) ( ) = − ρ ρ − λ ρ − − λ ρ − ln 1 I Tr Tr H E Tr 1 2 A A A A A A A A A δ = 0 I ρ = − β β exp( ) / ( ) H Z A A A [ ] [ ] − ρ ρ ≤ − ρ ρ ln ln Tr Tr A A A A If a stable Gibbs state exists, it attains the maximum of the von Neumann entropy with average energy fixed.
Unfortunately, the typicality argument cannot be applied to Schwarzschild BH evaporation! Actually, from our result, the typical state must be a Gibbs state, but…
No stable Gibbs state for Schwarzschild BH due to negative heat capacity! (Hawking –Page, 1983) 1 d E 1 = = = − 2 < E M 0 π BH π 8 GT 8 dT GT If there exists a stable Gibbs state, heat capacity must be positive. [ ] ( ) β = − β ( ) exp Z Tr H BH BH ( ) 2 − E E d E = > 0 2 dT T
Thus, a system of a black hole and Hawking radiation is not in typical states, at least in the sense of the Page curve hypothesis, during BH evaporation. Because we have no stable Gibbs state,“ thermal entropy” of Schwarzschild BH ( ) is not needed to be a upper /( G 4 ) A bound of entanglement entropy. ≤ ≈ /( G 4 ) S S A EE thermal
In ordinary quantum systems, U AB Ψ AB Microcanonical Energy Shell E total = . const Ψ is a typical state with almost certainty AB after a relaxation time.
The state of BH evaporation can be non-typical until the last burst. ⊗ U I BH HR Ψ Fast scrambling of BH BH + HR does not contribute to entanglement between BH and HR. ( emission ) U Non-chaotic HR emission generated by smooth space time curvature outside horizon Sub-Hilbert space of non-typical states
If so, how is the Page curve modified?
The moving mirror model is totally unitary. So we are able to learn how the information can be retrieved. The model is a tool to explore the Page curve hypothesis and its modification by using various mirror trajectories.
− + x x − ϕ ˆ ( ) x out mirror trajectory + = − ( ) x f x + ϕ ˆ ( ) x in
+ = Mirror ( ) − ( ) ± = ± x f x x t x Trajectory: ϕ = Boundary ˆ | 0 + = − Condition: ( ) x f x + − ( ) − ϕ = ϕ ϕ ˆ ˆ ˆ , ( ) ( ( )) x t x f x Solution: in in − = Scattering − ϕ ϕ ˆ ˆ ( ) ( ( )) x f x Relation: out in
ˆ = ∂ ϕ ∂ ϕ = ∂ ϕ ∂ ϕ ˆ ˆ ˆ ˆ : : : : T Out-going energy flux: − − − − − − out out 2 − − ∂ ∂ 3 2 1 ( ) 3 ( ) f x f x ˆ − = − − − − 0 ( ) 0 T x − − − − π ∂ ∂ in in 24 ( ) 2 ( ) f x f x − − − = − ϕ ϕ ˆ ˆ ( ) ( ( )) derived from x f x out in ( Dynamical Casimir Effect )
Moving Mirror Model in 1+1 dim. mimics 3+1 dim. spherical gravitational collapse. ( ) 1 − + − − κ = = − + x ( ) ln 1 x f x e κ − − ≈ −∞ ≈ ( ) f x x The mirror does not move in the past. ( ) 1 − − ≈ ∞ ≈ − − κ The mirror accelerates and ( ) κ exp f x x approaches the light trajectory, + = 0 . x acceleration
The mirror emits thermal flux in the late time. π − ˆ >> κ = 2 0 ( 1 / ) 0 T x T − − in in 12 κ acceleration = T Temperature: π 2 1 † ∝ ( ) ( ) out out ˆ ˆ 0 0 a a ω ω π in in 2 ω − exp 1 κ Hawking Radiation!
1+1 dim Moving Mirror Model as analogue of Hawking radiation emission κ = T π 2 − + x x Hawking Radiation Mirror Trajectory: [ ] ( ) + − = − + − κ ln 1 exp x x At rest → Gradually accelerated → Uniform acceleration
− + x x B Entanglement Entropy of Emitted Radiation − x 2 A − − ε x 2 B − − + ε x x 1 1 [ ] ρ = Ψ Ψ Tr mirror A B AB AB trajectory + = [ ] − = − ρ ln ρ ( ) x f x S Tr AB A A
Entanglement entropy: ( ) 2 − − − 1 ( ) ( ) f x f x = 2 1 ln S AB − − ε ∂ ∂ 2 12 ( ) ( ) f x f x − − 2 1 Lattice spacing (UV cutoff) Renormalized entanglement entropy: = − ( ) vac S S S AB ren AB ( ) 2 − − − 1 ( ) ( ) f x f x = 2 1 ln ( ) 2 − − − − 12 ∂ ∂ − ( ) ( ) f x f x x x − − 2 1 2 1 Holzhey-Larsen-Wilczek
BH Evaporation Trajectory − + x x Hawking Radiation Mirror Trajectory: ( ) − + − κ 1 exp x + = − ( ( ) ) ln x − + κ − 1 exp x h At rest → Uniform Acceleration → At rest
κ = = 1 , 500 h 0.010 y 0.009 0.008 0.007 − ( ) T x 0.006 − − 0.005 0.004 0.003 0.002 0.001 0.000 0 100 200 300 400 500 600 700 − x x
Page Curve for BH Evaporation Trajectory y 20 − κ = = = − 1 , 500 , 2 h x 15 1 ∆ S EE 10 5 − x 0 2 0 100 200 300 400 500 600 700 x Page time Thanks to Daniel Harlow
In order to reproduce the Page curve, very strange time evolution induced by nonlocality is required for the mirror trajectories! Φ BH + HR Ψ ⊗ 0 BH HR Old black hole Young black hole M M BH BH Quite different time schedules of information leakage for black holes with the same mass.
Possible modification Planck-energy last burst with a tiny amount of of the Page curve, information assuming local dynamics. + x − x λ >> κ Hawking Radiation Planck BH scale Quantum energy Gravity scale Mirror Trajectory: ( ) − + − κ 1 exp x + = − ( ( ) ) ln x − + λ − 1 exp x h
Modified Page Curve for BH Evaporation 50 − κ = λ = = = − y 1 , 100 , 500 , 2 h x 1 40 Contribution 30 ∆ S of zero-point fluctuation EE without energy cost for 20 information storage 10 − x 0 2 0 100 200 300 400 500 600 700 x All of the information comes “Page time” out at the end by zero-point fluctuation flow .
Information Retrieval without Energy at the End + x Zero-Point Hawking Fluctuation Particle − x Quantum Gravity at the End Mirror Entangled Pair Trajectory in Vacuum State The entangled partner of the Hawking particle is zero-point fluctuation with zero energy. (Wilczek, Hotta-Schützhold-Unruh)
Entangled Partner Particle B Particle A Entanglement Hawking Zero-Point Fluctuation Flow Particle with Zero Energy (Wilczek, Hotta-Schützhold-Unruh, cf. Hawking-Perry-Strominger)
Therefore, the information loss problem may not be so serious, because small (or zero) amount of energy is enough to carry huge amount of quantum information in principle.
Black Hole Evaporation Zero-point fluctuation Hawking Radiation Apparent horizon 0 in Zero-point fluctuation Collapsing = 0 r Shell
Strong Subadditivity “Paradox” Late radiation B Early radiation C A Strong subadditivy: ≥ + − S S S S AB B ABC BC = , = 0 S S S No Drama: BC ABC A ≥ + S S S AB B A S > Page Curve S A AB Hypothesis
Strong Subadditivity “Paradox” Late radiation B Early radiation C A S > Page Curve S Hypothesis A AB > ≥ + S S S S A AB B A > 0 S B
Strong Subadditivity “Paradox” Late radiation B Early radiation C A Remnant& Zero-Point Fluctuation Flow S < S A AB until the last burst. Thus, no strong subadditivity paradox!
We don’t care the no drama condition breaks at the last burst, because the horizon is affected by quantum gravity. 50 y 40 30 ∆ S EE 20 > = 0 S S + A AB zr pt flc 10 − x 0 2 0 100 200 300 400 500 600 700 x The last burst
Summary ○ Adopting canonical typicality for nondegenerate systems with nonvanishing Hamiltonians, the entanglement becomes non-maximal, and BH firewalls do not emerge. ○ Typical states must be Gibbs states for smaller quantum systems. If we have stable Gibbs states for old Schwarzschild BH’s (and small AdS BH’s), the heat capacity must be positive. Because it is actually negative, the states of BH evaporation are not typical. ⇒ Inevitable Modification of the Page Curve Note: for a large AdS BH and Hawking radiation in a thermal equilibrium, the entanglement entropy equals the thermal entropy of the smaller system.
IV. Quantum Measurement of BH Firewalls “Another Firewall Paradox” M. Hotta, J. Matsumoto and K. Funo, Phys. Rev. D89, 124023 (2014)
The strong subadditivity paradox has been resolved. Free-fall observers do not encounter firewalls when come across event horizon in an average meaning. Classical Horizon Infalling Large BH Particle
However, we have another possibility of firewall emergence. The point is Reeh-Schlieder theorem in quantum field theory.
Reeh-Schlieder theorem: The set of states generated from by the polynomial 0 in algebra of local operators in any bounded spacetime region is dense in the total Hilbert space of the field. Thus, in principle, any state can be arbitrarily closely reproduced by acting a polynomial of local operators of E’ on . 0 in ∑∫ ∀ Ψ ≈ ˆ ˆ n ( , , ) ( ) ( ) 0 a x x O x O x d x 1 1 1 n n n n in n ' E ∑∫ ˆ ˆ n ( , , ) ( ) ( ) a x x O x O x d x 1 1 1 n n n n n ' E E’ L’ ⇒ Ψ 0 in
L Note that the Reeh-Schlieder property is f maintained in the time evolution: E in → 0 . f ˆ O E Even in the future infinity, we may remotely generate any excitation ' L with some probability smaller than 1. 0 ˆ O in ' E E ∑∫ ' E ˆ ˆ ˆ = n ( , , ) ( ) ( ) O a x x O x O x d x 1 1 1 n n n n n E T − − ˆ ˆ O O E f E − x L E
Firewall Measurement Paradox: x = − x fw L Imagine that, besides the background Hawking radiation, a wave packet i with positive energy of the order of L ψ the radiation temperature appears i E − x = at . Then the firewall (FW) x E fw ? FW + x = appears at . ( ) g h x fw + x = ( ) g h x fw ˆ ˆ ∝ M O If measurement operator is constructed from iE E Reeh-Schlieder operation, an arbitrary post- measurement state including firewalls can ∑ = ψ emerge. f i i E L i Firewall! Measured
V. Informational Cosmic Censorship Conjecture “Resolution of the Paradox from a viewpoint of Quantum Measurement Energy Cost” M. Hotta, J. Matsumoto and K. Funo, Phys. Rev. D89, 124023 (2014)
L Because the mirror merely stretches the modes of the field, the future measurement i L E is equivalent to a past measurement for the in- vacuum state. i ' L ' L ˆ ⇔ ˆ M M ' iE iE ˆ M We can analyze the problem ' iE ' E using past infinity.
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